Something I had come up with long ago, but most certainly observed by others two hundred years earlier: Let z be a primitive (n-th, think exp(2*i*Pi/n)) root in the cyclic group of a finite field. Set cos( 2*Pi/n ) := (z^2 + 1) / (2*z) i * sin( 2*Pi/n ) := (z^2 - 1) / (2*z) (check that exp = cos + i * sin, also that cos^2 + sin^2 = 1) This is useful for number theoretic transforms, cf. Fxtbook page 808. Here i must be some 4th root of 1, so finite fields GF( p^e ) for p=4k+3 and odd e seem to be out. Still, one can do the computations in some extension field, notably GF( p^2 ) for GF( p ) for p = 4k+3. I have never thought about what those "angles" might be good for, though. I also have omitted GF( 2^e ), what could we do for those? Best regards, jj * Fred Lunnon <fred.lunnon@gmail.com> [Sep 06. 2017 15:44]:
Nice ideas in the article, albeit marred by notational confusion.
More generally, we can define logarithms and quadrics over |F_p ; so what's to stop us defining angle and distance (modulo choice of exponential function)?
If so, how much of elementary Euclidean geometry carries over --- eg. do similar triangles have proportional sides? Does Pythagoras' theorem hold?
Presumably somebody (Wildberger?) has investigated such questions; but I have not encountered them previously.
WFL
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